Demagnifying gravitational lenses toward hunting a clue of exotic matter and energy

0.32 MB·English

by  Takao Kitamura

Demagnifying gravitational lenses toward hunting a clue of exotic
matter and energy
Takao Kitamura, Koki Nakajima, and Hideki Asada
Faculty of Science and Technology,
Hirosaki University, Hirosaki 036-8561, Japan
(Dated: January 17, 2013)
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Abstract
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2
We examine a gravitational lens model inspired by modified gravity theories and exotic matter
n
a
J and energy. We study an asymptotically flat, static, and spherically symmetric spacetime that is

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modified in such a way that the spacetime metric depends on the inverse distance to the power
1

] of positive n in the weak-field approximation. It is shown analytically and numerically that there
c
q
is a lower limit on the source angular displacement from the lens object to get demagnification.
-
r
g Demagnifying gravitational lenses could appear, provided the source position β and the power
[

2 n satisfy β > 2/(n + 1) in the units of the Einstein ring radius under a large-n approximation.
v
9 Unusually, thetotal amplification of thelensed images, thoughtheyarecausedby thegravitational
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3 pull, could be less than unity. Therefore, time-symmetric demagnification parts in numerical light
0
1. curves by gravitational microlensing (F.Abe, Astrophys. J. 725, 787, 2010) may be evidence of
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2 an Ellis wormhole (being an example of traversable wormholes), but they do not always prove it.
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: Such a gravitational demagnification of the light might be used for hunting a clue of exotic matter
v
i
X and energy that are described by an equation of state more general than the Ellis wormhole case.
r
a Numericalcalculations forthen = 3and10casesshowmaximally 10and 60percentdepletion
∼ ∼
of the light, when the source position is β 1.1 and β 0.7, respectively.
∼ ∼
PACS numbers: 04.40.-b,95.30.Sf, 98.62.Sb
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I. INTRODUCTION
The bending of light was the first experimental confirmation of the theory of general
relativity. Nowadays, gravitational lensing is one of the most important tools in astronomy
and cosmology. It is widely used for investigating extrasolar planets, dark matter, and dark
energy.
Light bending is also of theoretical importance, in particular for studying a null structure
of a spacetime. A rigorous form of the bending angle plays an important role in properly
understanding a strong gravitational field [1–5]. For example, strong gravitational lensing
in a Schwarzschild black hole was considered by Frittelli, Kling, and Newman [1] and by
Virbhadra and Ellis [2]; Virbhadra and Ellis [3] later described strong gravitational lensing
by naked singularities; Eiroa, Romero, and Torres [4] treated Reissner-Nordstr¨om black hole
lensing; Perlick [5] discussed lensing by a Barriola-Vilenkin monopole and also by an Ellis
wormhole.
One of the peculiar features of general relativity is that the theory admits a nontrivial
topologyofa spacetime; forinstance, awormhole. AnEllis wormholeis aparticular example
of the Morris-Thorne traversable wormhole class [6–8]. Many years ago, scattering problems
in such spacetimes were discussed (for instance, [9, 10]). One remarkable feature is that
the Ellis wormhole has zero mass at the spatial infinity, but it causes light deflection [9,
10]. Moreover, gravitational lensing by wormholes has been recently investigated as an
observational probe of such an exotic spacetime [5, 11–16]. Several forms of the deflection
angle by the Ellis wormhole have been recently derived and often used [5, 13–18]. A reason
for such differences has been clarified [19, 20].
According to recent numerical calculations by Abe [14], time-symmetric demagnification
parts in light curves could appear by gravitational microlensing effects of the Ellis wormhole.
Is thetime-symmetric demagnification evidence forthe Ellis wormhole? Isis very interesting
to address this question. One reason is that wormholes are inevitably related with viola-
tions of some energy conditions in physics [21]. For instance, dark energy is introduced to
explain the observed accelerated expansion of the universe by means of anadditional energy-
momentum component in the right-hand side of the Einstein equation. Furthermore, the
left-hand side of the Einstein equation, equivalently the Einstein-Hilbert action, could be
modified in various ways (nonlinear curvature terms, higher dimensions, and so on) inspired
2
by string theory, loop quantum gravity, and so on. Because of the nonlinear nature of grav-
ity, modifications to one (or both) side(s) of the Einstein equation might admit spacetimes
significantly different from the standard Schwarzschild spacetime metric, even if the space-
time is assumed to be asymptotically flat, static, and spherically symmetric. One example
is an Ellis wormhole (being an example of traversable wormholes).
Inspired by a huge number of modified theories, this brief paper assumes, in a phe-
nomenological sense, that an asymptotically flat, static, and spherically symmetric modified
spacetime could depend on the inverse distance to the power of positive n in the weak-field
approximation. The Schwarzschild spacetime and the Ellis wormhole correspond to n = 1
and n = 2, respectively. Note that Birkhoff’s theorem could say that cases n = 1 might
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be nonvacuum, if the models were interpreted in the framework of the standard Einstein
equation.
The slightly modified gravitational lensing in modified gravity theories– such as a fourth
order f(R) gravity theory– has attracted interests (e.g., Refs. [22–24]). It has been shown
that the total magnification of the lensed images is stable and always larger than unity
against a small spherical perturbation of the Schwarzschild lens [25]. This suggests that
demagnifying gravitational lenses would need a significantly modified structure of the space-
time. The main purpose of this paper is to discuss demagnifying gravitational lenses due to
significantly modified spacetimes.
We take the units of G = c = 1 throughout this paper.
II. MODIFIED SPACETIME MODEL AND MODIFIED DEFLECTION ANGLE
OF LIGHT
This paper assumes that an asymptotically flat, static, and spherically symmetric modi-
fied spacetime could depend on the inverse distance to the power of positive n in the weak-
field approximation. We consider light propagating through a four-dimensional spacetime,
though the whole spacetime may be higher-dimensional. The four-dimensional spacetime
metric is expressed as
ε ε
2 1 2 2 2 2 2 2 2 2 2
ds = 1 dt + 1+ dr +r (dθ +sin θdφ )+O(ε ,ε ,ε ε ), (1)
−(cid:16) − rn(cid:17) (cid:16) rn(cid:17) 1 2 1 2
3
where r is the circumference radius and ε and ε are small bookkeeping parameters in
1 2
the following iterative calculations. Here, ε and ε may be either positive or negative,
1 2
respectively. A negative ε and ε for n = 1 corresponds to a negative mass (in the linearized
1 2
Schwarzschild metric).
For investigating light propagation, it is useful below to make a conformal transformation
with a factor of (1 ε /rn)1/2. The null structure (such as the light propagation) is not
1
−
affected by the conformal transformation. At the linear order of ε and ε , the spacetime
1 2
metric takes a simpler form:
ε
2 2 2 2 2 2 2 2
ds¯ = dt + 1+ dR +R (dθ +sin θdφ )+O(ε ), (2)
− (cid:16) Rn(cid:17)
where ε nε +ε and
1 2
≡ r2
2
R . (3)
ε
≡ 1
1
(cid:16) − rn(cid:17)
Note that only one parameter ε enters the conformally transformed metric.
For this metric, one can find the Lagrangian for a massless particle. Without loss of
generality, we focus on the equatorial plane θ = π/2, since the spacetime is spherically
symmetric. By using the constants of motion associated with the timelike and rotational
Killing vectors, the deflection angle of light is calculated at the linear order as
∞
dφ(R)
α = 2 dR π
Z dR −
R0
π
ε 2
= cosnψdψ +O(ε2), (4)
bn Z
0
where R and b denote the closest approach and the impact parameter of the light ray,
0
respectively. This deflection angle recovers the Schwarzschild (n = 1) and Ellis wormhole
(n = 2) cases. For particular cases, the above (always positive) integral factor becomes
π
2 (n 1)!!π
cosnψdψ = − (evenn),
Z n!! 2
0
(n 1)!!
= − (oddn),
n!!
√πΓ n+1
= 2 (realn > 0), (5)
2 Γ(cid:0)n+2(cid:1)
2
(cid:0) (cid:1)
Henceforth, thedeflection angleis denoted simply asα(b) = ε¯/bn by absorbing the numerical
constant into the ε¯parameter.
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III. MODIFIED LENS EQUATION AND ITS SOLUTIONS
Under the thin-lens approximation, it is useful to consider the lens equation as [26]
b D
LS
β = α(b), (6)
D − D
L S
where β denotes the angular position of the source and D , D , D are the distances from
L S LS
the observer to the lens, from the observer to the source, and from the lens to the source,
respectively. Wewish to consider significant magnification (ordemagnification), which could
occur for a source in (or near) the Einstein ring. The Einstein ring is defined for β = 0
[26]. If ε < 0, Eq. (6) has no positive roots for β = 0 because of the repulsive force in
the particular gravity model. For ε > 0, on the other hand, there is always a positive root
corresponding to the Einstein ring. The negative ε case is of less astronomical relevance.
Therefore, let us consider the positive ε case (causing the gravitational pull) in the following.
In units of the Einstein ring radius, Eq. (6) is rewritten as
1
ˆ ˆ ˆ
β = θ (θ > 0), (7)
− θˆn
1
ˆ ˆ ˆ
β = θ+ (θ < 0), (8)
( θˆ)n
−
ˆ ˆ
where β β/θ and θ θ/θ for the angular position of the image θ b/D .
E E L
≡ ≡ ≡
Let us consider two lines defined by Y = 1/θˆn and Y = θˆ β in the θˆ Y plane. For
− −
ˆ
θ > 0, therefore, we have only one intersection of the two lines that corresponds to one
ˆ
image position. Similarly, only one image appears for θ < 0.
For a general positive n (e.g., n = 5), it is impossible to find exact solutions for the
modified lens equation. To clarify the parameter dependence, we employ analytic but ap-
proximate methods rather than numerical calculations. Furthermore, in astronomy only the
significantly amplified images become detectable in gravitational microlensing. Such events
occur only when a source such as a distant star crosses the Einstein ring. We thus focus on
ˆ
such an Einstein ring-crossing case as β < 1 in units of the Einstein ring, for which Eqs. (7)
ˆ
and (8) are solved in the Taylor series form with respect to β. We obtain
1 1 n
ˆ ˆ ˆ2 ˆ3 ˆ
θ = 1+ β + β +O(β ) (θ > 0), (9)
+ n+1 2(n+1)2
1 1 n
ˆ ˆ ˆ2 ˆ3 ˆ
θ− = 1+ β β +O(β ) (θ < 0). (10)
− n+1 − 2(n+1)2
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IV. DEMAGNIFICATION CONDITION
The amplification factor denoted as A is (β/θ)(dβ/dθ) −1, namely, the inverse Jacobian
| |
of the gravitational lens mapping between the source and image position vectors [26]. By
using Eqs. (9) and (10), the amplification factor of each image, which is denoted by A and
+
A−, respectively, becomes
1
ˆ0
A± = +O(β ), (11)
ˆ
β(n+1)
ˆ
whereadifferencebetweenA+ andA− appearsatthenextorderinβ. Thetotalamplification
is thus
Atot A+ +A−
≡
2
ˆ0
= +O(β ). (12)
ˆ
β(n+1)
ˆ
For the Schwarzschild case (n = 1), A = 1/β. This is always larger than unity for
tot
ˆ
β < 1, in concordance with the well-known fact. Demagnification of the total lensed images
could occur, however, if
2
ˆ
β > . (13)
n+1
The larger the power n, the more likely the demagnification. One might guess that demag-
ˆ ˆ
nification could be caused for a smaller β, especially β = 0. However, this is not the case.
ˆ
Equation (13) suggests that the total demagnification could occur only when β is small but
larger than the critical value 2/(n+1) under a large-n approximation. Note that the com-
ˆ
patibility of the assumption β < 1 and Eq. (13) implies n > 1. Namely, Eq. (13) becomes
a better approximation as n grows larger than unity.
ˆ
The above argument is based on the near-zone approximation (β < 1). For a test of the
analytic result, we perform numerical calculations. We consider n = 10, which might be
one of the higher-dimensional models inspired by string theory. Equation (13) suggests that
ˆ
demagnification of the total lensed images could occur only for β > 2/11 = 0.182. Figure 1
shows numerical results for n = 1,2,3, and 10. In the case of n = 10, the analytic result for
ˆ ˆ
the critical value β = 2/11 = 0.182 is in good agreement with the numerical one, β = 0.187.
Figure 2 shows numerical light curves for n = 1,2,3, and 10. As the power n is larger,
time-symmetric demagnification parts in the light curves become longer in time and larger
in depth. Cases of n = 3 and 10 show maximally 10 and 60 percent depletion of the
∼ ∼
ˆ ˆ
light, when the source position is β 1.1 and β 0.7, respectively.
∼ ∼
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FIG. 1: Total amplification factor of the lensed images as a function of the source position βˆ
for n = 1,2,3, and 10. Top left, top right, bottom left, and bottom right panels correspond to
n = 1,2,3, and 10, respectively. In the case of n = 10, the total amplification factor is larger than
unity for βˆ< 0.187, whereas it is smaller for βˆ> 0.187. For convenience, a thin (red) line denotes
Atot = 1.
Before closing this section, we briefly mention an effective mass. A simple application
of the standard lens theory [26] suggests that the deflection (α = ε¯/bn) and magnification
studied here correspond to a convergence (scaled surface-mass density) of the form
ε¯(1 n) 1
κ(b) = − . (14)
2 bn+1
For n > 1, therefore, the effective surface-mass density of the lens object is interpreted as
negative in the framework of the standard lens theory. This means that the matter (and
energy) need to be exotic if n > 1.
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FIG. 2: Numerical light curves for the same minimum impact parameter of the light trajectory
βˆ0 = 0.1. The source star moves at constant speed and the source position changes as βˆ(t) =
(βˆ2 + t2)1/2, where time is normalized by the Einstein ring radius crossing time. Top left, top
0
right, bottom left, and bottom right panels correspond to n = 1,2,3, and 10, respectively. For
convenience, a thin (red) line denotes Atot = 1.
V. DISCUSSION AND CONCLUSION
We examined a gravitational lens model inspired by modified gravity theories and exotic
matter and energy. By using an asymptotically flat, static, and spherically symmetric space-
time model of which the metric depends on the inverse distance to the power of positive
n, it was shown in the weak-field and thin-lens approximations that demagnifying gravita-
ˆ
tional lenses could appear, provided the impact parameter of light β and the power n satisfy
ˆ
β > 2/(n+1) in units of the Einstein ring radius under a large-n approximation.
Therefore, time-symmetric demagnification parts in numerical light curves by gravita-
tional microlensing may be evidence of an Ellis wormhole, but they do not always prove
8
it. Such a gravitational demagnification of light might be used for hunting a clue of exotic
matter and energy that are described by an equation of state more general than the Ellis
wormhole case. Examples of n = 3 and 10 show maximally 10 and 60 percent depletion
∼ ∼
ˆ ˆ
of the light, when the source position is β 1.1 and β 0.7, respectively. It is left to future
∼ ∼
work to perform a numerical campaign for the vast parameter space.
The above gravitational demagnification of light occurs presumably because modified
lenses couldact asaneffectively negative (quasilocal) mass onaparticular light ray(through
Ricci focusing). Regarding this issue, a more rigorous formulation is needed. It would be
interesting tostudyarelationbetweenthemodel parameternandvitalmodifiedgravitythe-
ories (or matter models with an exotic equation of state) and also to make an interpretation
of the parameter n in the framework of the theory of general relativity.
ˆ
The analytical approximate solution in this paper is obtained at the linear order of 1/β
to discuss the total magnification. Tsukamoto and Harada [27] have studied the next or-
ˆ
der of 1/β to discuss the signed magnification sums, namely the difference between the
amplifications of two images.
We would like to thank F. Abe, T. Harada, S. Hayward, K. Nakao, and M. Visser for
useful conversations. We wish to thank N. Tsukamoto for stimulating comments on the
lecture (T.K.) on the present subject at “WormHole Workshop 2012” at Rikkyo University.
[1] S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D 61, 064021 (2000).
[2] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000).
[3] K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 65, 103004 (2002).
[4] E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D 66, 024010 (2002).
[5] V. Perlick, Phys. Rev. D 69, 064017 (2004).
[6] H. G. Ellis, J. Math. Phys. 14, 104 (1973).
[7] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
[8] M. S. Morris K. S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988).
[9] L. Chetouani and G. Cl´ement, Gen. Relativ. Gravit. 16, 111 (1984).
[10] G. Cl´ement, Int. J. Theor. Phys. 23, 335 (1984).
[11] M. Safonova, D. F. Torres, and G. E. Romero, Phys. Rev. D 65, 023001 (2001).
9
[12] A. A. Shatskii, Astron. Rep. 48, 525 (2004).
[13] K. K. Nandi, Y. Z. Zhang, and A. V. Zakharov, Phys. Rev. D 74, 024020 (2006).
[14] F. Abe, Astrophys. J. 725, 787 (2010).
[15] Y. Toki, T. Kitamura, H. Asada, and F. Abe, Astrophys. J. 740, 121 (2011).
[16] N. Tsukamoto, T. Harada, K. Yajima, Phys. Rev. D 86, 104062 (2012).
[17] T. K. Dey and S. Sen, Mod. Phys. Lett. A 23, 953 (2008).
[18] A. Bhattacharya and A. A. Potapov, Mod. Phys. Lett. A 25, 2399 (2010).
[19] K. Nakajima and H. Asada, Phys. Rev. D 85, 107501 (2012).
[20] G. W. Gibbons and M. Vyska, Classical Quantum Gravity 29, 065016 (2012).
[21] M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP, New York, 1995).
[22] S. Capozziello, V. F. Cardone, and A. Troisi, Phys. Rev. D 73, 104019 (2006).
[23] Z. Horvath, L. A. Gergely, D. Hobill, S. Capozziello, and M. De Laurentis, arXiv:1207.1823.
[24] S. Mendoza, T. Bernal, X. Hernandez, J. C. Hidalgo, and L. A. Torres, arXiv:1208.6241.
[25] H. Asada, Prog. Theor. Phys. 125, 403 (2011).
[26] P. Schneider, J. Ehlers, and E. E. Falco, Gravitational Lenses (Springer, NerYork, 1992).
[27] N. Tsukamoto and T. Harada, arXiv:1211.0380.
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